Strangely it is also not possible anymore to create new projects - the wizard appears but just restarts when I press create.
As I found out the paths for Visual Studio are now built from settings in the registry. Namely
HKEY_CURRENT_USER\Software\Microsoft\VisualStudio.
Comparing a colleagues installation with mine revealed no different settings.
So this is how the Property Pages/Configuration Properties/VC++ Directories look like:

I can only guess that this behavior was caused by a reinstallation of Studio a couple of months ago (to a different folder). As we use an external build-script for our main project there is a good chance that it is broken since then.

Are you sure you installed all the required features? cl.exe is the Visual Studio C/C++ compiler & linker. Make sure you added VC++ support, and not just C# and the other languages.
–
BreakthroughOct 5 '11 at 11:17

C++ support is added. I doubt that I would see the C++ templates if it was not installed.
–
tunerOct 6 '11 at 7:27

None of this helped so I went for a repair installation of VS 2010 and SP1 - it didn't help. After that I reset the settings - didn't help.
What got me going again is opening Property Pages of my project, then Configuration Properties/General/Platform Toolset. Here are three options listed in my installation:

v100

v90

Windows7.1SDK

The active one was v100 so I switched to v90. With that toolset VS is able to compile again. Obviously this is not a good solution because the 9 compiler is not the latest compiler so this is a mere workaround for my problem.

Project creation issue

The repair install for VS didn't help here either. This stackoverflow question solved a part of the problem. After adding .vbproj and vcproj to HKEY_CURRENT_USER\Software\Classes in the registry, I was able to create Visual Basic and C# projects again. But still I can't create any C++ projects.

Because of the drawbacks of the solution I will not mark my answer as solution to the problem. To me it looks like the registry is messed up and I would appreciate to get to know the source of my problems.